Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι be given.
Apply L5 with
7fe8f.. x0 x1 x2.
Assume H6:
and (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0) (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x3 (x1 x4 x5) = x1 (x1 x3 x4) x5).
Assume H7:
∃ x3 . and (prim1 x3 x0) (and (∀ x4 . prim1 x4 x0 ⟶ and (x1 x3 x4 = x4) (x1 x4 x3 = x4)) (∀ x4 . prim1 x4 x0 ⟶ ∃ x5 . and (prim1 x5 x0) (and (x1 x4 x5 = x3) (x1 x5 x4 = x3)))).
Apply H6 with
7fe8f.. x0 x1 x2.
Assume H9:
∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x3 (x1 x4 x5) = x1 (x1 x3 x4) x5.
Apply andI with
4f2b4.. (987b2.. x2 x1),
Subq x2 x0 leaving 2 subgoals.
Apply unknownprop_18724cb817cb481b8967b1efe13d39d42daa75ec771890da7ad3a6b6981daf0e with
x2,
x1.
Apply and3I with
∀ x3 . prim1 x3 x2 ⟶ ∀ x4 . prim1 x4 x2 ⟶ prim1 (x1 x3 x4) x2,
∀ x3 . prim1 x3 x2 ⟶ ∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ x1 x3 (x1 x4 x5) = x1 (x1 x3 x4) x5,
∃ x3 . and (prim1 x3 x2) (and (∀ x4 . prim1 x4 x2 ⟶ and (x1 x3 x4 = x4) (x1 x4 x3 = x4)) (∀ x4 . prim1 x4 x2 ⟶ ∃ x5 . and (prim1 x5 x2) (and (x1 x4 x5 = x3) (x1 x5 x4 = x3)))) leaving 3 subgoals.
The subproof is completed by applying H4.
Let x3 of type ι be given.
Let x4 of type ι be given.